Wigner function and pair production in parallel electric and magnetic fields

We derive analytical formulas for the equal-time Wigner function in an electromagnetic field of arbitrary strength. While the magnetic field is assumed to be constant, the electric field is assumed to be space-independent and oriented parallel to the magnetic field. The Wigner function is first decomposed in terms of the so-called Dirac-Heisenberg-Wigner (DHW) functions and then the transverse-momentum dependence is separated using a new set of basis functions which depend on the quantum number $n$ of the Landau levels. Equations for the coefficients are derived and then solved for the case of a constant electric field. The pair-production rate for each Landau level is calculated. In the case of finite temperature and chemical potential, the pair-production rate is suppressed by Pauli's exclusion principle.


I. INTRODUCTION
Quantum electrodynamics (QED) of strong electromagnetic (EM) fields has been studied for a very long time [1]. In the initial stage of non-central heavy-ion collisions, the electromagnetic field can be as large as 9.8 × 10 22 V/m at the Relativistic Heavy Ion Collider (RHIC) [2] and even larger at the Large Hadron Collider (LHC). Such extremely strong fields are generated by the fast-moving nuclei but will rapidly fall off with time [3]. The medium was estimated to extend the lifetime of the fields and enhance the possibility of detecting the influence of strong EM fields [4,5].
A strong magnetic field leads to interesting effects related to the chiral anomaly of quantum chromodynamics (QCD). On the other hand, a strong electric field can lead to decay of the QED vacuum. When the field strength is near or above the critical strength E c = m 2 c 3 /q [6][7][8], where m is the mass of the particle and q its electric charge, particle-antiparticle pairs can be created from vacuum. This process is commonly called Schwinger process in honor of Julian Schwinger, who derived the pair-production rate in a famous work [8]. The rate is exponentially suppressed below the critical field strength, which is about 1.32 × 10 18 V/m for electron-positron production. Pair production is a nonlinear phenomenon and the corresponding experiment is important for studying QED beyond the perturbative regime. The mechanism can occur in many systems such as in the early Universe, around neutron stars, and in heavy-ion collisions, while it is expected to appear in strong-laser experiments like the free-electron X-ray laser XFEL [9,10] and the extreme-light infrastructure ELI [11].
Although the Schwinger process has been studied for more than half a century, calculating the pair production in an arbitrary electromagnetic field is still a challenging problem. The case of a vanishing magnetic field B(t, x) = 0 and a space-independent electric field has been exhaustively discussed, where the problem can be translated into solving the famous Vlasov equation of quantum kinetic theory [12][13][14]. It can be analytically solved for a constant electric field E(t) = E 0 and the Sauter-type field E(t) = E 0 sech 2 (t/τ ). Many theoretical methods are developed to deal with these two cases and go beyond these analytical benchmarks, such as directly through quantum field theory [8], WKB methods [15][16][17], instanton methods [18][19][20], the Wigner-function method [21][22][23], the numerical worldline loop method [24,25], and holographic methods [26][27][28]. In principle, some methods such as the Wigner-function method [21][22][23] can be applied to very general cases, but one faces a system of non-linear partial differential equations. However, the field configurations in cosmology or in heavy-ion collisions are much more complicated than the above mentioned cases. One might find an approximate solution by partitioning space-time into small cells and applying the analytical results for a constant electric field in each cell. This, however, may generate uncontrollable uncertainties because an instanton study [29] showed that temporal inhomogeneities tend to enhance the pair production while spatial ones tend to suppress it. Especially in heavy-ion collisions, where the EM fields vary rapidly in both space and time [5,30,31], a proper numerical treatment is necessary [32,33].
Nowadays many researchers are focusing on the Schwinger process in strong-laser experiments [34,35]. The critical field strength E c for e + e − pair production corresponds to an average laser intensity I c = 1 2µ0c E 2 c 2.3 × 10 29 W/cm 2 . Unfortunately, such a large intensity is difficult to generate in an experiment. In the ELI project [11], the laser pulse can only reach ∼ 10 26 W/cm 2 , which is three magnitudes lower than I c . The pair production in such a case is strongly suppressed by a factor exp(−πE c /E) 10 −66 . Clearly, the critical intensity I c is not attainable for laser experiments in the near future. Meanwhile, the electric field in heavy-ion collisions can reach eE ∼ m 2 π c 3 / eE c at RHIC [2], which provides realistic conditions to study pair production. In the recently discovered Dirac semimetals, massless Dirac fermions can be excited by an external electromagnetic field and may be experimentally observed through their transport properties. The production rate remains finite even if the Dirac fermions are massless [36], which is different from the Schwinger process in vacuum. According to Maxwell's equations, a varying electric field will generate a magnetic field. Analytical calculations show that a magnetic field which is parallel to the electric field can increase the pair-production rate [16,[37][38][39][40].
Recently the enhancement of the pair-production rate due to parallel magnetic fields has been studied in string theory [41][42][43]. The pair production rate is modified by the thermal medium [44][45][46]. In this paper, we will reproduce these results via the Wigner-function method. On the other hand, the pair production in parallel electromagnetic fields is related to the chiral anomaly [47,48] and to pseudoscalar condensation [49][50][51], which can be verified using the results of this paper. We will focus on these effects in future work. This paper is organized as follows: In Sec. II we will briefly introduce the equal-time Wigner function and the Dirac-Heisenberg-Wigner (DHW) functions. General equations of motion for the DHW functions are also listed in this section. In Sec. III we simplify the equations of motion for the DHW functions in a spatially homogeneous electric field and give analytical solutions for a constant electric field. A constant magnetic field, which is parallel to the electric field, is taken into account in Sec. IV. The DHW functions reflect the behavior of the Landau levels. Analytical solutions are derived when both electric and magnetic fields are constant. In Sec. V we read off the pairproduction rate from the DHW functions derived in Sec. IV. In Sec. VI we give a summary and provide an outlook to future work. Details about the auxiliary functions used in this paper and their properties are summarized in App. A.
We take fermions to have positive unit charge q = +e and the electric and magnetic fields to point in the zdirection. We use the following notations for four-vectors: X = (x µ ) = (t, r) = (t, x T , z) = (t, x, y, z) and P = (p µ ) = (E, p) = (E, p T , p z ) = (E, p x , p y , p z ). We also use the following differential operators Our units are natural Heaviside-Lorentz units, = c = k B = 0 = µ 0 = 1. The metric tensor is g µν = diag(+, −, −, −).

II. DHW FUNCTIONS AND THEIR EQUATIONS OF MOTION
In this section we define the DHW functions as expansion coefficients of the equal-time Wigner function. The choice of the gauge potential is to some degree arbitrary. Here we use the temporal gauge A 0 = 0, for which the EM fields are given by E = −∂ t A and B = ∇ × A. In principle the EM fields include contributions from external fields and contributions from all charged particles. But in this paper we will focus on the case of an external field only and neglect the interaction between particles, which corresponds to a free Fermi gas.
The gauge-invariant Wigner operator is given bŷ where ψ is the Dirac field operator for spin-1/2 particles. This formula represents the Fourier transform with respect to the relative position Y of the direct product of two fermion field operators at space-time points X + Y 2 and X − Y 2 , respectively. The gauge link between these two points renders the Wigner operator gauge-invariant and is defined as where A µ is the gauge potential, e.g. in this paper the electromagnetic potential. Taking the expectation value of the Wigner operator in a state |Ω , we obtain the Wigner function The Wigner function, defined in eight-dimensional phase space (x µ , p µ ), is Lorentz covariant but does not have a clear physical interpretation [52,53]. By integrating over the energy p 0 we obtain the corresponding equal-time Wigner function [54][55][56], which can be interpreted as a quasi-probability distribution in six-dimensional phase space (x, p) at time t. Such a procedure evidently breaks the Lorentz covariance, however, the equation of motion might simplify to that of an initial-value problem. On the other hand, we adopt the Hartree approximation, i.e., we treat the quantum EM field as a semi-classical EM field, from which we derive the following formula for the equal-time Wigner function W (t, x, p), The Hartree approximation ignores higher-loop radiative corrections and is a good approximation for strong EM fields. The equation of motion for the equal-time Wigner function can be derived from the Dirac equation. We consider a non-zero chemical potential µ associated with the conservation of fermion number, which, for the sake of simplicity, we assume to be constant in space-time, hence its derivatives vanish. An effective way to include the chemical potential is by adding a term +µN to the Dirac-Hamilton operatorĤ D , whereN is the fermion-number operator. The corresponding Dirac equation reads Taking the time derivative of Eq. (4) and simplifying the result using the Dirac equation, we obtain the following equation of motion [21]: where the operators D t , D x , and Π are generalized operators for time and spatial derivatives, as well as momentum, in the presence of an EM field, For spatially homogeneous EM fields F µν (t, x) = F µν (t), these operators become local, It can be easily checked that the equal-time Wigner function W (t, x, p) satisfies W † = γ 0 W γ 0 and can be decomposed in terms of the 16 independent generators of the Clifford algebra Γ i = {1, iγ 5 , γ µ , γ 5 γ µ , 1 2 σ µν }, where σ µν = i 2 [γ µ , γ ν ] is the anti-symmetric spin tensor. These 16 functions, commonly called Dirac-Heisenberg-Wigner (DHW) functions, are real functions of time t and six-dimensional phase space (x, p). The tensor part can be further decomposed into two vector functions Some of these DHW functions have a clear physical meaning [54], e.g. F determines the mass density, V µ the vectorcharge current density, A µ the chiral-charge current density, and S the magnetic-moment density. Substituting Eq. (9) into the equation of motion (6) and projecting onto the 16 basis matrices, we find a system of partial differential equations (PDEs) for the DHW functions where the DHW functions have been divided into four groups and each group is composed of four functions [57], In Eq. (11), we have introduced the two matrices where Π and D x were already defined in Eq. (7). For any three-dimensional column vector V, V T is the corresponding transposed vector (line vector) and V × represents the anti-symmetric 3 × 3 matrix the elements of which are V × ij = − ijk V k . The differential equations 11 are equivalent to the ones in Refs. [21,55] but here we write them in a matrix form. When dealing with the Landau levels in a constant magnetic field, this matrix form allows for more compact formulas [57].

III. SPATIALLY HOMOGENEOUS ELECTRIC FIELD
In this section we will simplify the equations of motion (11) for the DHW functions in a spatially homogeneous electric field and then give the solution for a constant electric field. The electric field is taken to point into the z-direction. In this case, the gauge potential is A similar procedure has been adopted in Ref. [21], where the authors only discussed the pair production in vacuum. In a thermal environment, the low-energy states are occupied, which blocks the production of pairs into these states. In this section, a thermal equilibrium distribution is assumed at the initial time. Since collisions between particles are not included, all existing particles are accelerated by the electric field and thus the distribution depends on the canonical momentum. We show that, in the solution, the thermal distribution appears as an overall suppression factor, which does not influence the structure of the PDE system. The basis used in this section is different from the one in Ref. [21], but both span the same Hilbert space and thus are equivalent to each other. The system of PDEs and corresponding initial conditions derived with the basis in this section provides a convenient framework to describe pair production in parallel electric and magnetic fields in Sec. IV.
Let us first consider the DHW functions for a free gas of fermions. These can be derived by first quantizing the field operators in terms of solutions for free particles, which can be found in any textbook of quantum field theory, and then inserting the field operators into the definition of the Wigner function. The result is here d s is the degeneracy of spin, which is d s = 2 for spin-1 2 particles, and is the Fermi-Dirac distribution for particles/anti-particles with energy E p and vector chemical potential µ, while β = T −1 is the inverse temperature. Note that the fermionic field operators in the definition (4) are not normalordered, therefore taking the expectation value in the state |Ω yields an additional ∓1, which appears in the square brackets in Eq. (15). Here V 0 , V, and F are the charge, current, and mass densities, respectively. All other DHW functions vanish for a free gas of fermions, P = A 0 = A = S = T = 0, which can be proven using the completeness relations for the Dirac spinors u(k, s) and v(k, s).
We now proceed to solve the equations of motion (11) for the DHW functions. Due to the absence of a magnetic field and translation invariance of the system, we can set the spatial derivative D x to zero and Π ≡ p. The matrices in Eq. (13) then simplify to and D t = ∂ t + eE(t)∂ pz . Then, the 16 equations of motion for the DHW functions can be divided into several groups. The equation for the charge density separates from the others and reads After integrating over the momentum p and neglecting the boundary terms (because there is no particle with infinite p z ), the above equation is nothing but the conservation of net charge. Furthermore, the ten equations of motion for the DHW functions F, V, A, and T decouple from the other five for the functions P, A 0 , and S. These latter ones will no longer be considered, because their initial values are zero and thus they will remain zero for later times as well. In matrix form we have where w(t, p) = (F, V, A, T ) T is a ten-dimensional vector consisting of ten DHW functions and M (p) is a 10 × 10 matrix Inspired by the form (15) of the free DHW functions, we make the following ansatz for the solution of Eq. (19), Here is the difference of the gauge potentials at time t and at initial time t 0 . The distribution thus depends on the canonical momentum, which reflects the acceleration of fermions in an electric field. Since acting the operator D t on p + eδA(t)e z gives zero, the term in the curly brackets in Eq. (21) behaves like a constant overall factor and can be taken out of Eq. (19). The value of this term is in the range (−1, 0), which is the effect of Pauli blocking by particles already present in the thermal system. Note that, since the matrix w(t, p) has dimension ten, in principle we would need ten basis vectors e i in the ansatz (21). However, we actually only need three because these form a closed sub-space under the operators D t and M (p), while the initial conditions are also inside this sub-space. These basis vectors are which are independent of t and p z , so that D t e i = 0 for all i = 1, 2, 3. Here we have introduced the transverse mass m T ≡ m 2 + p 2 T , so that the three basis vectors are properly normalized, e i · e j = δ ij . We can also check that they are closed under the operator M (p), Inserting the ansatz (21) into Eq. (19) and using Eq. (23) we can derive the equations of motion for the coefficient functions In order to solve this system of PDEs, we need to specify the initial condition. Here we choose the values of the DHW functions in the absence of an electric field. For an integrable electric field, which vanishes sufficiently rapidly for t → ±∞, such as the Sauter-type field E(t) = E 0 cosh −2 (t/τ ), we specify the initial condition for t 0 → −∞, where we take the DHW functions to assume the values given by Eq. (15). However, for a constant electric field E(t) = E 0 , the momentum shift will be infinitely large if we take t 0 → −∞, because a constant field is not integrable. In reality, fermions will collide with each other, kinetic energy will be converted to thermal energy, and the system will approach thermodynamical equilibrium. Here we make the assumption that the system is already in thermodynamical equilibrium at initial time t 0 . We should find a solution that coincides with Eq. (15) when the field strength is sufficiently small, E 0 → 0, i.e., The pair-production rate and the corresponding Wigner function have analytical solutions for both a constant field and a Sauter-type field, see Ref. [21] for details of the derivation from quantum kinetic theory, which we will not repeat here. In a constant field E(t) = E 0 the solutions do not depend on space-time coordinates, which is obvious because of translation invariance, where η ≡

IV. PARALLEL AND SPATIALLY HOMOGENEOUS ELECTRIC AND MAGNETIC FIELDS
In this section, we will consider spatially homogeneous electric and magnetic fields which are parallel to each other. Without loss of generality, the fields are assumed to point into the z-direction. We also assume the magnetic field to be constant in time. Then the solution can be simplified by separately considering the different Landau levels. We provide an analytical solution for the case when the electric field is also constant in time.

A. Initial conditions
Analogously to the case without magnetic field, we choose the DHW functions in a pure magnetic field as initial condition for the system (11) of PDEs. Since we consider this field to be constant in space and time, an analytical solution can be found. The corresponding covariant DHW functions in eight-dimensional phase space have been determined in Ref. [57]. In this paper, we set the axial chemical potential to zero, i.e., we do not consider the chiral magnetic effect. Then, using the results of Ref. [57] the covariant DHW functions read where Here, E (n) pz = m 2 + p 2 z + 2neB is the energy of the nth Landau level in a constant magnetic field and f F D is the Fermi-Dirac distribution function. The basis vectors e (n) i are given in Eq. (A4) of the appendix. Since the pair production is a dynamical process, it is more convenient to use the equal-time formula. We emphasize that the covariant DHW functions can be obtained from the equal-time ones by applying an additional Fourier transformation in t, i.e., t → p 0 , and conversely, the equal-time DHW functions can be derived from the covariant ones by integrating over p 0 . Here we give the equal-time DHW functions, Here, C pz V (n) (p 0 , p z ) and C (n) 2 (p z ) ≡ dp 0 (p 0 + µ)V (n) (p 0 , p z ), respectively. The p 0 -integrals can be performed, yielding the result The Fermi-Dirac distributions are the number densities in coordinate space for fermions/anti-fermions. The prefactor 2 − δ n0 is the spin degeneracy of the various Landau levels. Comparing with Eq. (15) without the electromagnetic field, Eq. (29) has more non-vanishing components and depends on the Landau levels n. We will show later that in a constant magnetic field, different Landau levels evolve independently.

B. Equations of Motion
In the presence of a constant magnetic field, the operator for the generalized spatial differentiation, cf. second Eq. (8), becomes D x = eB × ∇ p , where the ordinary spatial gradient ∇ x has been dropped, since all considered fields are spatially homogeneous and the system is translation-invariant.
The lowest Landau level is special since we only need the basis vector e For the higher Landau levels, the situation is more complicated. In the last subsection we have shown that the basis vectors e (n) i , i = 1, 2, 3, cf. Eq. (A4), are necessary to describe the equal-time DHW functions in a constant magnetic field. One can easily check that these basis vectors are not closed under the operator M 2 defined in Eq. (13). In order to construct a closed space under M 2 , we need another basis vector, e (n) 4 , the definition of which is also given in Eq. (A4). Acting with M 1 , M 2 onto these basis vectors and using the relations (A9) gives for all higher Landau levels n > 0 where the coefficient matrices are Note that the transpose of these matrices appears in Eq. (32).
We have already seen in Eq. (29) that, when the electric field vanishes, the DHW functions can be expressed in terms of the basis vectors e (n) i . Taking Eq. (29) as initial condition one can straightforwardly conclude that the DHW functions will stay in the space spanned by e (n) i when they evolve according to the equation of motion (11). This is because D t acting on e (n) i gives zero, while we have already seen that these basis vectors form a closed subset when acting with M 1,2 onto them, see Eq. (32). We thus make the ansatz where i, j = 1, 2, 3, 4. Since the magnetic field is assumed to be constant in time, the basis vectors e  ij . For the lowest Landau level we obtain The equations for the higher levels are where f i4 ) T is a four-dimensional column vector. We observe that, on account of the orthogonality relations (A7), (A8), the equations for the different Landau levels separate from each other, which greatly facilitates the solution of the equations of motion.

C. Lowest Landau level
The spin of the fermion in the lowest Landau level with positive/negative charge is parallel/anti-parallel to the magnetic field. The equation for f (0) 2 , cf. the second line in Eq. (35), decouples from the other equations and gives rise to the conservation of net fermion number in the lowest Landau level. In order to see this, we note that the net fermion number density V 0 (t, p) is the first component of G 2 in Eq. (12). The lowest Landau level contributes just f where we have integrated by parts and neglected the boundary term. The equation This solution describes an overall acceleration of all charged particles. We note that in this paper we focus on a free fermion gas, so there are no collisions to prevent the acceleration. The equations of motion for the other three functions f 3,4 are coupled with each other. In order to simplify the problem, we make an ansatz which splits off the thermal distribution functions, where C (0) 1 is defined in Eq. (30). Here, p z + eA(t) = p z − eE 0 t is the canonical momentum. When acting with D t on f (0) i , we only need to consider its effect on χ i , because D t C (0) 1 (p z − eE 0 t) = 0. Thus, the equations of motion for the χ i are the same as the one for the corresponding f Comparing the ansatz (39) with the initial condition (29), i.e., for E 0 → 0, we find The system (40) of PDEs with the initial condition (41) coincides with the PDE system (24) in a pure electric field (substituting 3 and setting p 2 T = 0). One can therefore immediately give the solution for a constant electric field E(t) = E 0 , Inserting these functions into Eq. (34), one obtains the contribution from the lowest Landau level to the DHW functions.

D. Higher Landau levels
For the higher Landau levels (n > 0) we can read off from Eqs. (29), (34) that, when switching off the electric field, the only functions which do not vanish are f , each forms a closed set of homogeneous PDEs. However, since all of them have vanishing values when the electric field is zero, all of them will stay zero during the further evolution, even after switching on the electric field.
In the following, we therefore focus on the seven non-trivial functions f where p z − eE 0 t describes the overall acceleration of all existing particles by the electric field in the z-direction.
where d dt n (n) vac (p z ) is the pair-production rate in vacuum for given quantum numbers p z and n, The Fermi-Dirac distributions in the square bracket in Eq. (57) describe the suppression of pair production due to the Pauli exclusion principle. Summing Eq. (57) over all Landau levels yields the total pair-production rate.
In a medium where the chemical potential is zero but the temperature is nonzero the suppression factor is 1 − , which suppresses the production of pairs with small energies. This factor agrees with the result of Ref. [45]. However, the integral in Eq. (57) is hard to calculate numerically, because the auxiliary functions d 1 and d 2 are highly oscillatory at large q z , which makes the integration converge too slowly. This problem can be solved by separating the vacuum contribution from the thermal contribution. The pair-production rate in the nth Landau level in vacuum can be analytically calculated, using the asymptotic behavior of the d 1,2 functions, cf. App. A, The total rate from all Landau levels is which was previously derived in Refs. [37,38,40]. We see that the rate will be enhanced for B E 0 compared to that without the magnetic field [8]. Similarly we can also derive the production rate of chiral charge dn 5 /dt in a strong magnetic field, which gives the anomaly with the pair production [58,59].
The thermal contribution in Eq. (57) for the nth Landau level is The Fermi-Dirac distributions provide an exponential suppression ∼ e −E (n) qz for large q z , thus the q z -integral converges quickly. In order to show the thermal suppression in a physically intuitive way, we introduce the ratio r of the thermal to the vacuum contribution. This ratio is a function of time t and the three dimensionless parameters eẼ 0 ≡ eE0 [m (n) ] 2 , T ≡ T m (n) , andμ ≡ µ m (n) , where m (n) = √ m 2 + 2neB is the effective mass in the nth Landau level. The total pair-production rate in the nth Landau level is given by In order to show the thermal influence on pair production, we choose the time t = 0, which is when the canonical momentum equals the kinetic one. Figure 1 shows the function r(0,Ẽ 0 ,T ,μ) at finite dimensionless temperature and chemical potential. The values stay between −1 and 0 for all parameters considered, which describes the thermal suppression of pair production as demanded by the Pauli exclusion principle: a quantum state has a higher probability to be occupied at higher temperature or higher chemical potential; this occupation will block the production of new pairs with the same quantum numbers. When the electric field is strong enough, pairs with higher energies, which have smaller thermal occupation numbers, are more likely to be excited. Thus the suppression is inversely proportional to the electric field strength.

VI. SUMMARY
In this paper we have analytically calculated the Wigner function as well as the Schwinger pair production in constant and parallel electric and magnetic fields. We have derived the equation of motion for the equal-time Wigner function, whose sixteen components, the so-called DHW functions, have definite physical meanings. One can relate the Schwinger pair-production rate to some of these functions. For the case of a pure constant electric field, we took the vacuum values for the sixteen DHW functions as initial condition. Then, we obtained an analytic solution for the system of PDEs for the DHW functions. For parallel electric and magnetic fields, we adopted a similar method to calculate the DHW functions. We showed that the contributions of different Landau levels separate from each other. Under the replacement p 2 T → 2neB the system of PDEs and the condition when the electric field vanishes coincide with those in a pure electric field for each Landau level. This provides us with a new method for calculating the pair production in parallel electric and magnetic fields. Analytical solutions for the DHW functions for the case of constant electric and magnetic fields, together with the pair-production rate in each Landau level are derived. Our results can be directly generalised to the case of finite temperature and chemical potential. The calculation shows that the pair-production rate is thermally suppressed and the suppression is proportional to the thermodynamic variables T and µ. More energetic pairs can be created in a stronger electric field, which are less likely to be Pauli-blocked by the thermal distribution, and this leads to a decrease of the suppression factor.
The equation of motion for the Wigner function is equivalent to the Dirac equation if we adopt the classical-field approximation. However, the Wigner function contains sixteen independent components, which leads to a sixteendimensional system of PDEs. Due to advances in computer technology in the past few decades, it becomes possible to numerically solve this PDE system in some simplified cases. In this paper, we have found a set of basis functions in the presence of a constant magnetic field. These basis functions provide us with a way to replace the continuous transverse momenta p x , p y by the discrete Landau level index n. The parameter space of the Wigner function is then simplified from six-dimensional phase space (x, p) to the four-dimensional space spanned by (x, p z ) plus one discrete parameter n, which makes the system of PDEs more amenable for a numerical solution. However, the case considered in this paper, i.e., homogeneous and parallel electromagnetic fields, is effectively only a (1 + 1)-dimensional problem, whereas the fields in real experiments are more likely to be space-time dependent. Nevertheless, the way of decomposing the Wigner function presented here may inspire future works and may be a convenient starting point for the Wigner-function approach.